Let's Get Physical

Analog
synthesis can make interesting sounds, and sample-based synthesis has
its charms, but here's another approach: What if you were to analyze
the acoustic behavior of a few of your favorite instruments and then
express all of that information as a complex set of physics equations?
You'd have to consider the interaction of bow and string, reed and
wood, and mouthpiece and brass. You'd also have to account for every
conceivable performance technique that your favorite virtuosos have at
their disposal. Then, you would solve those equations in real time in
response to input from your favorite MIDI controller and sit back and
enjoy the beautiful sounds that result.

Insane? Not really. That is the aim of physical-modeling (PM)
synthesis, and though it is a bit of a miracle, it's actually not a new
concept. The first commercially available PM synthesizer was Yamaha's
VL1, which was introduced in 1993, and PM's roots date back more than a
century.

PM synthesis exists at the nexus of several fields of science. It
involves converting into physics equations the acoustician's
observations on what makes instruments tick. Mathematicians come up
with ingenious ways to solve those complex equations, and computer
engineers develop the hardware and software to crunch all of those
numbers quickly enough to respond immediately to a performer's input.
Speaking of input, PM's multiple-variable nature lends itself to the
intricacies of wind controllers and other imaginative MIDI devices,
bringing one more field of endeavor to the table.

In this column, I'll take a simplified look at the fundamental logic
of PM synthesis and explain why it's such a powerful and flexible
method. I will spare you the gory details of equations and such, so the
math-shy among you can relax.

GETTING EXCITED

To construct a mathematical model of an instrument's behavior,
conceptualize what occurs when the instrument is played. For example,
when a violin is bowed, the bow is dragged across a string, pulling it
away from its resting position until the string's tension overcomes the
bow's friction. At that point, the string slips back toward its resting
position until the bow's friction overcomes the string's tension and
starts pulling it again. This pull-and-slip cycle causes the string to
vibrate, inducing vibration in the instrument's body. That vibration is
the source of the violin's sound.

FIG. 1: This shows the four basic components of a physical model. The exciter initiates a vibration in the resonator. As the vibration passes through the instrument, it may encounter various impedances at points of transition. The shape and material of the instrument may also filter the vibration, dampening certain modes of vibration.

In PM terms, the bow scraping across the string is an
exciter, and the violin's body is called a resonator (see
Fig. 1). The exciter-resonator interaction is the core
ingredient in PM, and it's easy to see how the idea applies to other
instruments. A drum resonates in response to the excitation provided by
a stick or a mallet striking the head, and a guitar resonates when a
string is excited by a plectrum (a scientist would lose all credibility
by calling it a one-syllable name such as pick).

Any fiddle player knows there's more to a good violin sound than a
bow and a string, and your physical model continues with a look at the
rest of the instrument. When the string vibrates against the bridge, it
passes some of its energy on to the bridge and then to the violin body.
The bridge's density causes it to vibrate less readily than the string
— it has a greater impedance — so some vibration is
reflected back to the string instead of being passed on. For the same
reason, only part of the vibration is transmitted from bridge to body,
affecting the way the vibration propagates.

Last, but certainly not least, the body of the violin has its own
acoustic properties, and instead of merely amplifying the string's
sound, the body also acts as a filter, shaping the tone of the
string. The body's shape and size — along with the position,
size, and geometry of the f holes — determine which frequencies
are enhanced or attenuated, thereby determining the character of the
filtering effect.

A SAXY MODEL

Now I'll take a closer look at how the four components —
exciter, resonator, impedance, and filter — can be applied to
model a saxophone. The excitation is initiated by the player blowing
into the mouthpiece. That rush of air creates an area of lower pressure
along the top surface of the reed. The low pressure pulls the reed
toward the tip of the mouthpiece, restricting the airflow into the
mouthpiece and allowing the pressure difference between the top and
bottom surfaces of the reed to equalize. The tension in the reed from
being bent is then able to pull the reed back toward its original
position, where its tension is relieved and the airflow is great enough
to start the process all over again. This tug-of-war is the source of
the reed's vibration.

Therefore, the physical model of a saxophone starts by borrowing
equations from aerodynamics to evaluate the force bending the reed
toward the tip of the mouthpiece. Next, turn to structural engineering
to quantify how strongly the reed resists being bent. Now calculate the
amount of air that passes through the mouthpiece with each oscillation
of the reed, and you can evaluate the intensity of the excitation.

Consider the number of variables in the reed model even before it
interacts with the resonator. As the player blows harder, the reed
deflects farther before it snaps back and lets more air pass through
the mouthpiece, resulting in a more intense excitation. Sax players
know that reeds of different stiffnesses respond differently to a given
air pressure. (Saxophonists also realize that if you start trying to
model what makes a reed “good” or “bad,”
everyone will go mad.)

Clearly, the complexity of any musical instrument defies
comprehensive description. It's true that the more thoroughly you can
model an instrument, the more realistic your synthesized version will
be, but some degree of compromise is necessary. You can settle for a
less thorough model, or you can find shortcuts in the calculations. Of
course, limiting what behaviors you model can affect a virtual
instrument's sound and playability.

FEEDBACK FROM A SAX?

The reed's vibration excites the resonator, the column of air
contained within the body of the saxophone. As the player closes and
opens different combinations of keys, the saxophone effectively changes
length. The longer the column of air, the lower the fundamental
frequency at which it resonates.

Think back to the exciter for a moment. If the reed's vibration
excites the column of air to resonate but the resonant frequency of the
column changes with its length, shouldn't the vibration frequency of
the reed change correspondingly? After all, it doesn't make sense for
the exciter to vibrate at one frequency and the resonator to vibrate at
an unrelated frequency, does it?

FIG. 2: Physical models demonstrate either coupled or decoupled exciter-resonator pairs. A coupled exciter-resonator pair exhibits a feedback relationship, in which the resonator influences the behavior of the exciter. A decoupled pair follows a feedforward structure, with no such effect on the exciter.

In fact, the vibration of the air column is the primary factor in
determining the vibratory frequency of the reed. This is an example of
a feedback relationship, inasmuch as the resonator's response to
excitation manages to influence the behavior of the exciter (see
Fig. 2). This type of coupled exciter-resonator pair is
typical of wind instruments and bowed strings, in which an excitation
is continuous and the resonance is periodic.

In contrast, if you strike a drumhead with a stick, the head doesn't
feed any information back to the stick beyond bouncing it away. That is
called a feedforward relationship or a decoupled
exciter-resonator pair. Similarly, if you pluck a guitar string, the
string doesn't have any opportunity to influence the behavior of the
pick.

An additional aspect of the exciter-resonator relationship is that
resonators are generally linear in their behavior, whereas
exciters are generally nonlinear. If you apply more force to the
resonating air column inside the sax, it responds with proportionately
more output, or volume. A nonlinear exciter such as the saxophone reed,
however, reaches a point at which additional air pressure causes it to
close off completely, muting the sound, or to vibrate in an
unpredictable way, causing a squeak.

RESISTANCE IS FUTILE

Just as the intersection of a violin's bridge and strings presents
an impedance to the exciter's vibrations, so do changes in the diameter
of a saxophone's body (or any wind instrument). However, a saxophone's
conical shape results in an essentially infinite series of minute
impedance changes rather than the single large impedance presented by a
violin bridge (or a sudden, abrupt change in a wind instrument's
diameter). The tonal effect of those tiny continuous changes in
impedance is greater resonant support for even and odd harmonics. By
contrast, the clarinet is essentially cylindrical with a closed
mouthpiece and open bell, and its tonal spectrum consists primarily of
odd harmonics.

In addition, the bends in the neck and bell of a saxophone also
affect its tone. Calculating the impedance of a saxophone, therefore,
requires computing a continuously changing set of boundaries as well as
the effects of several curves in the tube.

FIG. 3: Applied Acoustics'' Tassman is a modular virtual synthesizer built on physical-modeling principles. The Player view (top) and Builder view each show the primary building blocks of a PM synth. Impedance effects are built into the plate and controlled by the Decay knob. Each module features modulation inputs, and various other parameters can be controlled by MIDI.

Once you've created the impedance equations, you can easily build an
alto sax that's straightened out like a soprano sax or even build a
straight bass sax if you wish, just by subtracting curves from the
model. You don't even have to worry about the physical constraints that
would make such an instrument unbelievably awkward to play. Among the
many attractions of PM is its ability to create those sorts of what-if
variations on instruments (see Fig. 3).

Any wind musician knows that the body of an instrument comes alive
with the sound of the notes it plays. Whether it's made of wood, brass,
or silver, it absorbs some of the energy of the vibrating column of
air, transmitting part of that energy to the surrounding air and
absorbing some with its own inertia. That filtering effect in the
saxophone model is computed by analyzing the energy transfer that
dampens the vibration of the air and contributes to the overall timbre
identified as a saxophone. At the same time, you must calculate the
filtering effects of the sound radiation pattern of the instrument's
bell and the rest of the body.

CHOPS TO SPARE

If all of this were only a way to create convincing synthesized
versions of acoustic instruments, that might be enough. After all,
people build elaborate multisampled instruments that Velocity-switch
from a hard to a soft attack. A PM instrument's equations allow its
attack to change continuously according to Velocity, a feat that would
require 128 Velocity layers in a sampler. Similarly, a PM synth doesn't
merely pitch-shift a sample to sound different pitches — it
recalculates the exact length of string or tube required to produce
that pitch, resulting in a consistent timbre and realistic transitions
from note to note.

As a saxophonist, I vary the lip pressure on the reed to bend the
pitch and create vibrato. A model of that behavior has only to factor
that change in reed position into the pressure/tension equation to
create a realistic response to the pressure of my lip on a
wind-controller mouthpiece. No more cheesy low-frequency oscillator
vibratos!

If a violinist bows nearer the bridge (sul ponticello) or
over the fingerboard (sur la touche), the tone changes
considerably. If your model properly represents the bow and string
relationship, that performance practice can be reproduced properly,
whereas a sampler would require two entirely new patches to be created.
Similarly, the effect of a trumpet mute can be modeled by modifying
impedance and filtering equations instead of starting a whole new
program from scratch. That flexibility permits PM to reproduce the wide
range of performance techniques employed by acoustic musicians in an
easily controlled way.

A physical model can, by its very design, reproduce some of the most
human effects of acoustic instruments, from the breathiness that
saxophones get at low volumes to the squeaks they produce when
overblown. The number of sample layers required to achieve that range
of expression is beyond the capabilities of even the best modern
samplers.

SONIC HYBRIDS

Some of the most interesting aspects of PM, however, arise when you
reach beyond the re-creation of existing instruments and apply a bit of
imagination. Have you ever wondered what it would sound like to bow a
bassoon? Once you've reduced exciter and resonator to their component
equations, who says you can't cross-pollinate them?

Why not build a trombone whose size decreases as its pitch rises? By
the time you get into the treble clef, you're playing something more
like a trumpet. Speaking of trumpets, the instrument used by Wynton
Marsalis is designed to be much more rigid than other trumpets, greatly
reducing the energy radiated from anywhere other than the bell for a
more intense sound. Why not test that theory by designing a PM trumpet
made entirely of granite?

By the time you get around to designing imaginary exciters and
resonators, you'll have a palette of amazing complexity and potential.
However, that complexity turns out to be PM's biggest limitation. The
“classical” approach I've described is elegant in its
conception, but it requires solving an enormous set of interrelated
equations to create each sound; in fact, it's often called the brute
force method. Even with the latest processors, you can't quite pull
off PM in that manner in real time.

As a result, several simplified approximations have been developed.
The most common approach is called waveguide synthesis, which is
employed in currently available PM synthesizers. Another approach is
called modal synthesis, which divides complex physical and tonal
structures into smaller, more manageable substructures. In any event,
PM synthesis opens a whole new world of sonic generation and
manipulation for electronic musicians to explore.

Brian Smithersis associate course director of MIDI at
Full Sail Real World Education in Winter Park, Florida. You can reach
him through his Web site athttp://members.aol.com/notebooks1.